divisor from $|\mathcal{L}^{\otimes 2}\otimes\mathcal{O}(2,3)|$ on the $\mathbb{P}^2$-bundle $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(-1,-1)^{\oplus 2})$ over $\mathbb{P}^1\times\mathbb{P}^1$ such that $X\cap Y$ is irreducible, and $\mathcal{L}$ is the tautological bundle, and $Y\in|\mathcal{L}|$

divisor from the linear system $|(\alpha\circ\pi_1)^*(\mathcal{O}_{\mathbb{P}^2}(1)\otimes\pi_2^*(\mathcal{O}_{\mathbb{P}^2}(2))|$ where $\pi_i\colon\mathrm{Bl}_1\times\mathbb{P}^2$ are the projections, and $\alpha\colon\mathrm{Bl}_1\mathbb{P}^2\to\mathbb{P}^2$ is the blowup